Synthesis of a quantum circuit

ABSTRACT

Systems, computer-implemented methods, and computer program products to facilitate synthesis of a quantum circuit are provided. According to an embodiment, a system can comprise a memory that stores computer executable components and a processor that executes the computer executable components stored in the memory. The computer executable components can comprise a circuit generation component that generates, iteratively, quantum circuits from 1 to N two-qubit gates, wherein at least one or more iterations ( 1, 2 , . . . , N) adds a single two-qubit gate to circuits from a previous iteration based on using added single 2-qubit gates that represent operations distinct from previous operations relative to previous iterations. The computer executable components can further comprise a circuit identification component that identifies, from the quantum circuits, a desired circuit that matches a quantum circuit representation.

BACKGROUND

The subject disclosure relates to quantum circuits, and morespecifically, to synthesis of a quantum circuit.

Quantum computing is generally the use of quantum-mechanical phenomenafor the purpose of performing computing and information processingfunctions. Quantum computing can be viewed in contrast to classicalcomputing, which generally operates on binary values with transistors.That is, while classical computers can operate on bit values that areeither 0 or 1, quantum computers operate on quantum bits (qubits) thatcomprise superpositions of both 0 and 1, can entangle multiple quantumbits, and use interference.

Quantum computing has the potential to solve problems that, due to theircomputational complexity, cannot be solved, either at all or for allpractical purposes, on a classical computer. A challenge in implementingquantum computing is efficient synthesis of certain quantum circuits(e.g., efficient synthesis in terms of computational costs). The size ofthe Clifford and controlled NOT-Dihedral (CNOT-Dihedral) groups growsexponentially with the number of qubits, so it is difficult to find anefficient synthesis of quantum circuits having such groups. It is alsodifficult to find an efficient synthesis (e.g., corresponding torelatively low computational costs) of such quantum circuits using aminimal number of physical basic gates (e.g., quantum gates), inparticular, 2-qubit gates such as, for instance, the controlled-X (CNOT)gate.

A problem with existing technologies that reduce the number of CNOTgates in a quantum circuit is that they do not reduce the number of CNOTgates in a Clifford quantum circuit having a Clifford group or aCNOT-Dihedral quantum circuit having a CNOT-Dihedral group. Anotherproblem with such existing technologies is that they involve aniterative constructive approach based on the number of qubits in aquantum circuit.

SUMMARY

The following presents a summary to provide a basic understanding of oneor more embodiments of the invention. This summary is not intended toidentify key or critical elements, or delineate any scope of theparticular embodiments or any scope of the claims. Its sole purpose isto present concepts in a simplified form as a prelude to the moredetailed description that is presented later. In one or more embodimentsdescribed herein, systems, devices, computer-implemented methods, and/orcomputer program products that facilitate synthesis of a quantum circuitare described.

According to an embodiment, a system can comprise a memory that storescomputer executable components and a processor that executes thecomputer executable components stored in the memory. The computerexecutable components can comprise a circuit generation component thatgenerates, iteratively, quantum circuits from 1 to N two-qubit gates,wherein at least one or more iterations (1, 2, . . . , N) adds a singletwo-qubit gate to circuits from a previous iteration based on usingadded single 2-qubit gates that represent operations distinct fromprevious operations relative to previous iterations. The computerexecutable components can further comprise a circuit identificationcomponent that identifies, from the quantum circuits, a desired circuitthat matches a quantum circuit representation. An advantage of such asystem is that it can minimize a number of CNOT gates in the desiredcircuit.

In some embodiments, the desired circuit comprises a defined number ofCNOT gates and a Clifford quantum circuit having a Clifford group or aCNOT-Dihedral quantum circuit having a CNOT-Dihedral group, and thecomputer executable components further comprise an application componentthat performs randomized benchmarking on a defined number of qubits in aquantum device based on the desired circuit, thereby facilitating atleast one of improved efficiency, improved performance, or reducedcomputational costs associated with at least one of the applicationcomponent, the processor, or the system in performing the randomizedbenchmarking based on the desired circuit. An advantage of such a systemis that it can minimize a number of CNOT gates in the desired circuit,thereby facilitating at least one of improved efficiency, improvedperformance, or reduced computational costs associated with at least oneof the application component, the processor, or the system in performingthe randomized benchmarking based on the desired circuit.

According to another embodiment, a computer-implemented method cancomprise generating, by a system operatively coupled to a processor,iteratively, quantum circuits from 1 to N two-qubit gates, wherein atleast one or more iterations (1, 2, . . . , N) adds a single two-qubitgate to circuits from a previous iteration based on using added single2-qubit gates that represent operations distinct from previousoperations relative to previous iterations. The computer-implementedmethod can further comprise identifying, by the system, from the quantumcircuits, a desired circuit that matches a quantum circuitrepresentation. An advantage of such a computer-implemented method isthat it can be implemented to minimize a number of CNOT gates in thedesired circuit.

In some embodiments, the desired circuit comprises a defined number ofCNOT gates and a Clifford quantum circuit having a Clifford group or aCNOT-Dihedral quantum circuit having a CNOT-Dihedral group, and theabove computer-implemented method can further comprise performing, bythe system, randomized benchmarking on a defined number of qubits in aquantum device based on the desired circuit, thereby facilitating atleast one of improved efficiency, improved performance, or reducedcomputational costs associated with at least one of the processor or thesystem in performing the randomized benchmarking based on the desiredcircuit. An advantage of such a computer-implemented method is that itcan be implemented to minimize a number of CNOT gates in the desiredcircuit, thereby facilitating at least one of improved efficiency,improved performance, or reduced computational costs associated with atleast one of the processor or the system in performing the randomizedbenchmarking based on the desired circuit.

According to another embodiment, a computer program product facilitatinga quantum circuit synthesis process is provided. The computer programproduct comprising a computer readable storage medium having programinstructions embodied therewith, the program instructions executable bya processor to cause the processor to generate, by the processor,iteratively, quantum circuits from 1 to N two-qubit gates, wherein atleast one or more iterations (1, 2, . . . , N) adds a single two-qubitgate to circuits from a previous iteration based on using added single2-qubit gates that represent operations distinct from previousoperations relative to previous iterations. The program instructions arefurther executable by the processor to cause the processor to identify,by the processor, from the quantum circuits, a desired circuit thatmatches a quantum circuit representation. An advantage of such acomputer program product is that it can minimize a number of CNOT gatesin the desired circuit.

In some embodiments, the desired circuit comprises a defined number ofCNOT gates and a Clifford quantum circuit having a Clifford group or aCNOT-Dihedral quantum circuit having a CNOT-Dihedral group, and theprogram instructions are further executable by the processor to causethe processor to perform, by the processor, randomized benchmarking on adefined number of qubits in a quantum device based on the desiredcircuit, thereby facilitating at least one of improved efficiency,improved performance, or reduced computational costs associated with theprocessor in performing the randomized benchmarking based on the desiredcircuit. An advantage of such a computer program product is that it canminimize a number of CNOT gates in the desired circuit, therebyfacilitating at least one of improved efficiency, improved performance,or reduced computational costs associated with the processor inperforming the randomized benchmarking based on the desired circuit.

According to an embodiment, a system can comprise a memory that storescomputer executable components and a processor that executes thecomputer executable components stored in the memory. The computerexecutable components can comprise a circuit generation component thatgenerates, during a first iteration, a first set of quantum circuitscomprising 2-qubit gates. The computer executable components can furthercomprise a second circuit generation component that generates, during asecond iteration, a second set of quantum circuits by adding a 2-qubitgate to the first set of quantum circuits such that the second set ofquantum circuits are selected to a redundant operation use the 2-qubitgate without introducing the redundant operation to that of the firstset of quantum circuits. The computer executable components can furthercomprise a circuit identification component that identifies, from thefirst set of quantum circuits and the second set of quantum circuits, adesired circuit that matches a quantum circuit representation. Anadvantage of such a system is that it can minimize a number of CNOTgates in the desired circuit.

In some embodiments, the desired circuit comprises a defined number ofCNOT gates and a Clifford quantum circuit having a Clifford group or aCNOT-Dihedral quantum circuit having a CNOT-Dihedral group, and thecomputer executable components further comprise an application componentthat performs randomized benchmarking on a defined number of qubits in aquantum device based on the desired circuit, thereby facilitating atleast one of improved efficiency, improved performance, or reducedcomputational costs associated with at least one of the applicationcomponent, the processor, or the system in performing the randomizedbenchmarking based on the desired circuit. An advantage of such a systemis that it can minimize a number of CNOT gates in the desired circuit,thereby facilitating at least one of improved efficiency, improvedperformance, or reduced computational costs associated with at least oneof the application component, the processor, or the system in performingthe randomized benchmarking based on the desired circuit.

According to another embodiment, a computer-implemented method cancomprise generating, by a system operatively coupled to a processor,during a first iteration, a first set of quantum circuits comprising2-qubit gates. The computer-implemented method can further comprisegenerating, by the system, during a second iteration, a second set ofquantum circuits by adding a 2-qubit gate to the first set of quantumcircuits such that the second set of quantum circuits are selected to aredundant operation use the 2-qubit gate without introducing theredundant operation to that of the first set of quantum circuits. Thecomputer-implemented method can further comprise identifying, by thesystem, from the first set of quantum circuits and the second set ofquantum circuits, a desired circuit that matches a quantum circuitrepresentation. An advantage of such a computer-implemented method isthat it can be implemented to minimize a number of CNOT gates in thedesired circuit.

In some embodiments, the desired circuit comprises a defined number ofCNOT gates and a Clifford quantum circuit having a Clifford group or aCNOT-Dihedral quantum circuit having a CNOT-Dihedral group, and theabove computer-implemented method can further comprise performing, bythe system, randomized benchmarking on a defined number of qubits in aquantum device based on the desired circuit, thereby facilitating atleast one of improved efficiency, improved performance, or reducedcomputational costs associated with at least one of the processor or thesystem in performing the randomized benchmarking based on the desiredcircuit. An advantage of such a computer-implemented method is that itcan be implemented to minimize a number of CNOT gates in the desiredcircuit, thereby facilitating at least one of improved efficiency,improved performance, or reduced computational costs associated with atleast one of the processor or the system in performing the randomizedbenchmarking based on the desired circuit.

DESCRIPTION OF THE DRAWINGS

FIGS. 1 and 2 illustrate block diagrams of example, non-limiting systemsthat can facilitate synthesis of a quantum circuit in accordance withone or more embodiments described herein.

FIGS. 3, 4, 5, 6, and 7 illustrate flow diagrams of example,non-limiting computer-implemented methods that can facilitate synthesisof a quantum circuit in accordance with one or more embodimentsdescribed herein.

FIG. 8 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can befacilitated.

FIG. 9 illustrates a block diagram of an example, non-limiting cloudcomputing environment in accordance with one or more embodiments of thesubject disclosure.

FIG. 10 illustrates a block diagram of example, non-limiting abstractionmodel layers in accordance with one or more embodiments of the subjectdisclosure.

DETAILED DESCRIPTION

The following detailed description is merely illustrative and is notintended to limit embodiments and/or application or uses of embodiments.Furthermore, there is no intention to be bound by any expressed orimplied information presented in the preceding Background or Summarysections, or in the Detailed Description section.

One or more embodiments are now described with reference to thedrawings, wherein like referenced numerals are used to refer to likeelements throughout. In the following description, for purposes ofexplanation, numerous specific details are set forth in order to providea more thorough understanding of the one or more embodiments. It isevident, however, in various cases, that the one or more embodiments canbe practiced without these specific details.

Given the problems described above with prior art technologies, thepresent disclosure can be implemented to produce a solution to theseproblems in the form of systems, computer-implemented methods, and/orcomputer program products that can: generate, iteratively, quantumcircuits from 1 to N two-qubit gates, wherein at least one or moreiterations (1, 2, . . . , N) adds a single two-qubit gate to circuitsfrom a previous iteration based on using added single 2-qubit gates thatrepresent operations distinct from previous operations relative toprevious iterations; and/or identify, from the quantum circuits, adesired circuit that matches a quantum circuit representation. Anadvantage of such systems, computer-implemented methods, and/or computerprogram products is that they can be implemented to minimize a number ofCNOT gates in the desired circuit.

In some embodiments, the present disclosure can be implemented toproduce a solution to the problems described above in the form ofsystems, computer-implemented methods, and/or computer program productsthat can perform randomized benchmarking on a defined number of qubitsin a quantum device based on the desired circuit, where the desiredcircuit comprises a defined number of CNOT gates and a Clifford quantumcircuit having a Clifford group or a CNOT-Dihedral quantum circuithaving a CNOT-Dihedral group, thereby facilitating at least one ofimproved efficiency, improved performance, or reduced computationalcosts associated with at least one of a processor or a system inperforming the randomized benchmarking based on the desired circuit. Anadvantage of such systems, computer-implemented methods, and/or computerprogram products is that they can be implemented to minimize a number ofCNOT gates in the desired circuit, thereby facilitating at least one ofimproved efficiency, improved performance, or reduced computationalcosts associated with at least one of a processor or a system inperforming the randomized benchmarking based on the desired circuit.

FIGS. 1 and 2 illustrate block diagrams of example, non-limiting systems100 and 200, respectively, that can facilitate synthesis of a quantumcircuit in accordance with one or more embodiments described herein.System 100 and/or system 200 can comprise a quantum circuit synthesissystem 102. As illustrated in the example embodiment depicted in FIG. 1, quantum circuit synthesis system 102 of system 100 can comprise amemory 104, a processor 106, an interface component 108, a circuitgeneration component 110, a circuit identification component 112, and/ora bus 114. As illustrated in the example embodiment depicted in FIG. 2 ,quantum circuit synthesis system 102 of system 200 can further comprisea second circuit generation component 202 and/or an applicationcomponent 204.

In some embodiments, quantum circuit synthesis system 102 can beassociated with a cloud computing environment. For example, quantumcircuit synthesis system 102 can be associated with cloud computingenvironment 950 described below with reference to FIG. 9 and/or one ormore functional abstraction layers described below with reference toFIG. 10 (e.g., hardware and software layer 1060, virtualization layer1070, management layer 1080, and/or workloads layer 1090).

Quantum circuit synthesis system 102 and/or components thereof (e.g.,interface component 108, circuit generation component 110, circuitidentification component 112, second circuit generation component 202,application component 204, etc.) can employ one or more computingresources of cloud computing environment 950 described below withreference to FIG. 9 and/or one or more functional abstraction layers(e.g., quantum software, etc.) described below with reference to FIG. 10to execute one or more operations in accordance with one or moreembodiments of the subject disclosure described herein. For example,cloud computing environment 950 and/or such one or more functionalabstraction layers can comprise one or more classical computing devices(e.g., classical computer, classical processor, virtual machine, server,etc.), quantum hardware, and/or quantum software (e.g., quantumcomputing device, quantum computer, quantum processor, quantum circuitsimulation software, superconducting circuit, etc.) that can be employedby quantum circuit synthesis system 102 and/or components thereof toexecute one or more operations in accordance with one or moreembodiments of the subject disclosure described herein. For instance,quantum circuit synthesis system 102 and/or components thereof canemploy such one or more classical and/or quantum computing resources toexecute one or more classical and/or quantum: mathematical function,calculation, and/or equation; computing and/or processing script;algorithm; model (e.g., artificial intelligence (AI) model, machinelearning (ML) model, etc.); and/or another operation in accordance withone or more embodiments of the subject disclosure described herein.

It is to be understood that although this disclosure includes a detaileddescription on cloud computing, implementation of the teachings recitedherein are not limited to a cloud computing environment. Rather,embodiments of the present invention are capable of being implemented inconjunction with any other type of computing environment now known orlater developed.

Cloud computing is a model of service delivery for enabling convenient,on-demand network access to a shared pool of configurable computingresources (e.g., networks, network bandwidth, servers, processing,memory, storage, applications, virtual machines, and services) that canbe rapidly provisioned and released with minimal management effort orinteraction with a provider of the service. This cloud model may includeat least five characteristics, at least three service models, and atleast four deployment models.

Characteristics are as follows:

On-demand self-service: a cloud consumer can unilaterally provisioncomputing capabilities, such as server time and network storage, asneeded automatically without requiring human interaction with theservice's provider.

Broad network access: capabilities are available over a network andaccessed through standard mechanisms that promote use by heterogeneousthin or thick client platforms (e.g., mobile phones, laptops, and PDAs).

Resource pooling: the provider's computing resources are pooled to servemultiple consumers using a multi-tenant model, with different physicaland virtual resources dynamically assigned and reassigned according todemand. There is a sense of location independence in that the consumergenerally has no control or knowledge over the exact location of theprovided resources but may be able to specify location at a higher levelof abstraction (e.g., country, state, or datacenter).

Rapid elasticity: capabilities can be rapidly and elasticallyprovisioned, in some cases automatically, to quickly scale out andrapidly released to quickly scale in. To the consumer, the capabilitiesavailable for provisioning often appear to be unlimited and can bepurchased in any quantity at any time.

Measured service: cloud systems automatically control and optimizeresource use by leveraging a metering capability at some level ofabstraction appropriate to the type of service (e.g., storage,processing, bandwidth, and active user accounts). Resource usage can bemonitored, controlled, and reported, providing transparency for both theprovider and consumer of the utilized service.

Service Models are as follows:

Software as a Service (SaaS): the capability provided to the consumer isto use the provider's applications running on a cloud infrastructure.The applications are accessible from various client devices through athin client interface such as a web browser (e.g., web-based e-mail).The consumer does not manage or control the underlying cloudinfrastructure including network, servers, operating systems, storage,or even individual application capabilities, with the possible exceptionof limited user-specific application configuration settings.

Platform as a Service (PaaS): the capability provided to the consumer isto deploy onto the cloud infrastructure consumer-created or acquiredapplications created using programming languages and tools supported bythe provider. The consumer does not manage or control the underlyingcloud infrastructure including networks, servers, operating systems, orstorage, but has control over the deployed applications and possiblyapplication hosting environment configurations.

Infrastructure as a Service (IaaS): the capability provided to theconsumer is to provision processing, storage, networks, and otherfundamental computing resources where the consumer is able to deploy andrun arbitrary software, which can include operating systems andapplications. The consumer does not manage or control the underlyingcloud infrastructure but has control over operating systems, storage,deployed applications, and possibly limited control of select networkingcomponents (e.g., host firewalls).

Deployment Models are as follows:

Private cloud: the cloud infrastructure is operated solely for anorganization. It may be managed by the organization or a third party andmay exist on-premises or off-premises.

Community cloud: the cloud infrastructure is shared by severalorganizations and supports a specific community that has shared concerns(e.g., mission, security requirements, policy, and complianceconsiderations). It may be managed by the organizations or a third partyand may exist on-premises or off-premises.

Public cloud: the cloud infrastructure is made available to the generalpublic or a large industry group and is owned by an organization sellingcloud services.

Hybrid cloud: the cloud infrastructure is a composition of two or moreclouds (private, community, or public) that remain unique entities butare bound together by standardized or proprietary technology thatenables data and application portability (e.g., cloud bursting forload-balancing between clouds).

A cloud computing environment is service oriented with a focus onstatelessness, low coupling, modularity, and semantic interoperability.At the heart of cloud computing is an infrastructure that includes anetwork of interconnected nodes.

Continuing now with the example embodiments illustrated in FIGS. 1 and 2. It should be appreciated that the embodiments of the subjectdisclosure depicted in various figures disclosed herein are forillustration only, and as such, the architecture of such embodiments arenot limited to the systems, devices, and/or components depicted therein.For example, in some embodiments, system 100, system 200, and/or quantumcircuit synthesis system 102 can further comprise various computerand/or computing-based elements described herein with reference tooperating environment 800 and FIG. 8 . In several embodiments, suchcomputer and/or computing-based elements can be used in connection withimplementing one or more of the systems, devices, components, and/orcomputer-implemented operations shown and described in connection withFIG. 1 , FIG. 2 , and/or other figures disclosed herein.

Memory 104 can store one or more computer and/or machine readable,writable, and/or executable components and/or instructions that, whenexecuted by processor 106 (e.g., a classical processor, a quantumprocessor, etc.), can facilitate performance of operations defined bythe executable component(s) and/or instruction(s). For example, memory104 can store computer and/or machine readable, writable, and/orexecutable components and/or instructions that, when executed byprocessor 106, can facilitate execution of the various functionsdescribed herein relating to quantum circuit synthesis system 102,interface component 108, circuit generation component 110, circuitidentification component 112, second circuit generation component 202,application component 204, and/or another component associated withquantum circuit synthesis system 102 as described herein with or withoutreference to the various figures of the subject disclosure.

Memory 104 can comprise volatile memory (e.g., random access memory(RAM), static RAM (SRAM), dynamic RAM (DRAM), etc.) and/or non-volatilememory (e.g., read only memory (ROM), programmable ROM (PROM),electrically programmable ROM (EPROM), electrically erasableprogrammable ROM (EEPROM), etc.) that can employ one or more memoryarchitectures. Further examples of memory 104 are described below withreference to system memory 816 and FIG. 8 . Such examples of memory 104can be employed to implement any embodiments of the subject disclosure.

Processor 106 can comprise one or more types of processors and/orelectronic circuitry (e.g., a classical processor, a quantum processor,etc.) that can implement one or more computer and/or machine readable,writable, and/or executable components and/or instructions that can bestored on memory 104. For example, processor 106 can perform variousoperations that can be specified by such computer and/or machinereadable, writable, and/or executable components and/or instructionsincluding, but not limited to, logic, control, input/output (I/O),arithmetic, and/or the like. In some embodiments, processor 106 cancomprise one or more central processing unit, multi-core processor,microprocessor, dual microprocessors, microcontroller, System on a Chip(SOC), array processor, vector processor, quantum processor, and/oranother type of processor. Further examples of processor 106 aredescribed below with reference to processing unit 814 and FIG. 8 . Suchexamples of processor 106 can be employed to implement any embodimentsof the subject disclosure.

Quantum circuit synthesis system 102, memory 104, processor 106,interface component 108, circuit generation component 110, circuitidentification component 112, second circuit generation component 202,application component 204, and/or another component of quantum circuitsynthesis system 102 as described herein can be communicatively,electrically, operatively, and/or optically coupled to one another via abus 114 to perform functions of system 100, system 200, quantum circuitsynthesis system 102, and/or any components coupled therewith. Bus 114can comprise one or more memory bus, memory controller, peripheral bus,external bus, local bus, a quantum bus, and/or another type of bus thatcan employ various bus architectures. Further examples of bus 114 aredescribed below with reference to system bus 818 and FIG. 8 . Suchexamples of bus 114 can be employed to implement any embodiments of thesubject disclosure.

Quantum circuit synthesis system 102 can comprise any type of component,machine, device, facility, apparatus, and/or instrument that comprises aprocessor and/or can be capable of effective and/or operativecommunication with a wired and/or wireless network. All such embodimentsare envisioned. For example, quantum circuit synthesis system 102 cancomprise a server device, a computing device, a general-purposecomputer, a special-purpose computer, a quantum computing device (e.g.,a quantum computer), a tablet computing device, a handheld device, aserver class computing machine and/or database, a laptop computer, anotebook computer, a desktop computer, a cell phone, a smart phone, aconsumer appliance and/or instrumentation, an industrial and/orcommercial device, a digital assistant, a multimedia Internet enabledphone, a multimedia players, and/or another type of device.

Quantum circuit synthesis system 102 can be coupled (e.g.,communicatively, electrically, operatively, optically, etc.) to one ormore external systems, sources, and/or devices (e.g., classical and/orquantum computing devices, communication devices, etc.) via a data cable(e.g., High-Definition Multimedia Interface (HDMI), recommended standard(RS) 232, Ethernet cable, etc.). In some embodiments, quantum circuitsynthesis system 102 can be coupled (e.g., communicatively,electrically, operatively, optically, etc.) to one or more externalsystems, sources, and/or devices (e.g., classical and/or quantumcomputing devices, communication devices, etc.) via a network.

In some embodiments, such a network can comprise wired and wirelessnetworks, including, but not limited to, a cellular network, a wide areanetwork (WAN) (e.g., the Internet) or a local area network (LAN). Forexample, quantum circuit synthesis system 102 can communicate with oneor more external systems, sources, and/or devices, for instance,computing devices (and vice versa) using virtually any desired wired orwireless technology, including but not limited to: wireless fidelity(Wi-Fi), global system for mobile communications (GSM), universal mobiletelecommunications system (UMTS), worldwide interoperability formicrowave access (WiMAX), enhanced general packet radio service(enhanced GPRS), third generation partnership project (3GPP) long termevolution (LTE), third generation partnership project 2 (3GPP2) ultramobile broadband (UMB), high speed packet access (HSPA), Zigbee andother 802.XX wireless technologies and/or legacy telecommunicationtechnologies, BLUETOOTH®, Session Initiation Protocol (SIP), ZIGBEE®,RF4CE protocol, WirelessHART protocol, 6LoWPAN (IPv6 over Low powerWireless Area Networks), Z-Wave, an ANT, an ultra-wideband (UWB)standard protocol, and/or other proprietary and non-proprietarycommunication protocols. In such an example, quantum circuit synthesissystem 102 can thus include hardware (e.g., a central processing unit(CPU), a transceiver, a decoder, quantum hardware, a quantum processor,etc.), software (e.g., a set of threads, a set of processes, software inexecution, quantum pulse schedule, quantum circuit, quantum gates, etc.)or a combination of hardware and software that facilitates communicatinginformation between quantum circuit synthesis system 102 and externalsystems, sources, and/or devices (e.g., computing devices, communicationdevices, etc.).

Quantum circuit synthesis system 102 can comprise one or more computerand/or machine readable, writable, and/or executable components and/orinstructions that, when executed by processor 106 (e.g., a classicalprocessor, a quantum processor, etc.), can facilitate performance ofoperations defined by such component(s) and/or instruction(s). Further,in numerous embodiments, any component associated with quantum circuitsynthesis system 102, as described herein with or without reference tothe various figures of the subject disclosure, can comprise one or morecomputer and/or machine readable, writable, and/or executable componentsand/or instructions that, when executed by processor 106, can facilitateperformance of operations defined by such component(s) and/orinstruction(s). For example, interface component 108, circuit generationcomponent 110, circuit identification component 112, second circuitgeneration component 202, application component 204, and/or any othercomponents associated with quantum circuit synthesis system 102 asdisclosed herein (e.g., communicatively, electronically, operatively,and/or optically coupled with and/or employed by quantum circuitsynthesis system 102), can comprise such computer and/or machinereadable, writable, and/or executable component(s) and/orinstruction(s). Consequently, according to numerous embodiments, quantumcircuit synthesis system 102 and/or any components associated therewithas disclosed herein, can employ processor 106 to execute such computerand/or machine readable, writable, and/or executable component(s) and/orinstruction(s) to facilitate performance of one or more operationsdescribed herein with reference to quantum circuit synthesis system 102and/or any such components associated therewith.

Quantum circuit synthesis system 102 can facilitate (e.g., via processor106) performance of operations executed by and/or associated with one ormore components thereof (e.g., interface component 108, circuitgeneration component 110, circuit identification component 112, secondcircuit generation component 202, application component 204, etc.). Forexample, as described in detail below, quantum circuit synthesis system102 can facilitate (e.g., via processor 106): receiving a quantumcircuit representation; generating, iteratively, quantum circuits from 1to N two-qubit gates, where at least one or more iterations (1, 2, . . ., N) adds a single two-qubit gate to circuits from a previous iterationbased on using added single 2-qubit gates that represent operationsdistinct from previous operations relative to previous iterations;and/or identifying from the quantum circuits, a desired circuit thatmatches the quantum circuit representation. In this example, asdescribed in detail below, quantum circuit synthesis system 102 canfurther facilitate (e.g., via processor 106): generating, iteratively,the quantum circuits to minimize a number of CNOT gates in the desiredcircuit; and/or performing randomized benchmarking on a defined numberof qubits in a quantum device based on the desired circuit, where thedesired circuit comprises a defined number of CNOT gates and a Cliffordquantum circuit having a Clifford group or a CNOT-Dihedral quantumcircuit having a CNOT-Dihedral group, thereby facilitating at least oneof improved efficiency, improved performance, or reduced computationalcosts associated with at least one of a processor or a system inperforming the randomized benchmarking based on the desired circuit. Inthis example, the quantum circuits can comprise a Clifford quantumcircuit having a Clifford group and/or the quantum circuits can comprisea CNOT-Dihedral quantum circuit having a CNOT-Dihedral group.

In another example, as described in detail below, quantum circuitsynthesis system 102 can facilitate (e.g., via processor 106): receivinga quantum circuit representation; generating, during a first iteration,a first set of quantum circuits comprising 2-qubit gates; generating,during a second iteration, a second set of quantum circuits by adding a2-qubit gate to the first set of quantum circuits such that the secondset of quantum circuits are selected to a redundant operation use theadded single 2-qubit gate without introducing the redundant operation tothat of the first set of quantum circuits; and/or identifying from thefirst set of quantum circuits and the second set of quantum circuits, adesired circuit that matches the quantum circuit representation. In thisexample, as described in detail below, quantum circuit synthesis system102 can further facilitate (e.g., via processor 106): generating,iteratively, the first set of quantum circuits and the second set ofquantum circuits to minimize a number of CNOT gates in the desiredcircuit; and/or performing randomized benchmarking on a defined numberof qubits in a quantum device based on the desired circuit, where thedesired circuit comprises a defined number of CNOT gates and a Cliffordquantum circuit having a Clifford group or a CNOT-Dihedral quantumcircuit having a CNOT-Dihedral group, thereby facilitating at least oneof improved efficiency, improved performance, or reduced computationalcosts associated with at least one of a processor or a system inperforming the randomized benchmarking based on the desired circuit. Inthis example, the first set of quantum circuits and the second set ofquantum circuits can comprise Clifford quantum circuits having aClifford group and/or the first set of quantum circuits and the secondset of quantum circuits can comprise CNOT-Dihedral quantum circuitshaving a CNOT-Dihedral group.

To facilitate performance of one or more of such operations describedabove in accordance with one or more embodiments of the subjectdisclosure, quantum circuit synthesis system 102 (e.g., via processor106, interface component 108, circuit generation component 110, circuitidentification component 112, second circuit generation component 202,application component 204, etc.) can derive and/or implement the one ormore new algorithms and/or the one or more new lemmas described below.As referenced herein, the terms below can be defined as follows.

The n-qubit Clifford group C is the group of quantum circuits generatedby the gates H (Hadamard), S (phase), and CX (controlled-X), up to aglobal phase, where:

H[[1,1],[1,−1]] is the 1-qubit Hadamard gate (up to a global phase);

S[[1,0],[0,i]] is the 1-qubit phase gate;

V=SH (or any other equivalent 1-qubit Clifford element of order 3);

CX=[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] is the 2-qubit controlled-Xgate; and

C(r) is the subset of operators in C implementable by a circuit with rCX gates (and any number of 1-qubit gates).

The non-Clifford CNOT-Dihedral group G on n-qubits is generated by thegates X, T=T(m), and CX, up to a global phase, where:

X=[[0,1],[1,0]] is the 1-qubit Pauli-X gate;

Fix an integer m and define the 1-qubit gate T(m)=[[1,0],[0,u(m)]],where u(m)=e^(2πi/m) is an m-th root of unity (or any other equivalent1-qubit CNOT-Dihedral element of order m);

T gate can be denoted as T=T(m), although the T gate is usually definedas T(8); and

G(r) is the subset of operators in G implementable by a circuit with rCX gates (and any number of 1-qubit gates).

Algorithm 1: Successive Generation of a Clifford Circuit

C(r) is the subset of operators in the Clifford group C implementable bya circuit with r CX gates.

C(r+1) can be successively constructed from C(r) using Lemma 1 definedbelow.

Lemma 1:

Based on an assumption that U is any element in C(r+1), then U=V_(i)^(k)V_(j) ^(l)CX_(i,j)U′ for some 0≤i<j<n, 0≤k,l<2 and U′ in C(r), whereU′ is a corresponding element in C(r) that satisfies the equalityU=V_(i) ^(k)V_(j) ^(l)CX_(i,j)U′ as defined above.

In particular, |C(r+1)|≤9(n²−n)/2|C(r)|.

This bound is sharp: if n=2, then |C(1)|=9|C(0)|.

Algorithm 2: Successive Generation of a CNOT-Dihedral Circuit

G(r) is the subset of operators in the CNOT-Dihedral group Gimplementable by a circuit with r CX gates.

G(r+1) can be successively constructed from G(r) using Lemma 2 definedbelow.

Lemma 2:

Based on an assumption that U is any element in G(r+1), thenU=I_(i)T_(j) ^(l)CX_(i,j)U′ for some 0≤i,j<n, 0≤l<m/d, d=gcd(m,2) and U′in G(r), where U′ is a corresponding element in G(r) that satisfies theequality U=I_(i)T_(j) ^(l)CX_(i,j)U′ as defined above.

In particular, |G(r+1)|≤m(n²−n)/d|G(r)|.

This bound is sharp: if n=2, then |G(1)|=2m/d|G(0)|.

To facilitate implementation of one or more of the algorithms and/orlemmas defined above, in some embodiments described herein, interfacecomponent 108 (e.g., an application programming interface (API), arepresentational state transfer (REST) API, a graphical user interface(GUI), etc.) can receive a quantum circuit representation 116 (e.g., amathematical representation, an algebraic representation, etc.). Inthese embodiments, circuit generation component 110 can implement (e.g.,via processor 106) one or more of the algorithms and/or lemmas definedabove to generate, iteratively, quantum circuits 118 (also referred toherein as circuits 118) from 1 to N two-qubit gates 120, wherein atleast one or more iterations 122 (e.g., 1, 2, . . . , N, where N denotesa total quantity) adds a single two-qubit gate 124 (e.g., controlled-X(CNOT) gate, CNOT gate, etc.) to circuits 118 from a previous iteration126 based on using added single 2-qubit gates 124 (e.g., controlled-X(CNOT) gates, CNOT gates, etc.) that represent operations 128 distinctfrom previous operations 142 relative to previous iterations 126. Inthese embodiments, circuit identification component 112 can implement(e.g., processor 106) one or more of the algorithms and/or lemmasdefined above to identify, from quantum circuits 118, a desired circuit130 that matches quantum circuit representation 116. For example, inthese embodiments, circuit identification component 112 can identify,from quantum circuits 118, such a desired circuit 130 that matchesquantum circuit representation 116, where desired circuit 130 cancomprise a defined number of CNOT gates 132. For instance, in theseembodiments, the added single 2-qubit gates 124 described above cancomprise CNOT gates 132 that represent operations 128 distinct fromprevious operations 142 (e.g., operations 128 and previous operations142 are not redundant operations), and as such, desired circuit 130 cancomprise the least number (e.g., smallest number) of CNOT gates 132(e.g., the least number of CNOT gates 132 relative to the number of CNOTgates 132 in other successively generated quantum circuits 118).

In some embodiments, quantum circuits 118 described above can comprise aClifford quantum circuit 134 having a Clifford group 136. In theseembodiments, circuit generation component 110 can implement (e.g., viaprocessor 106) algorithm 1 and/or lemma 1 defined above to successivelygenerate the Clifford quantum circuits 134 having a Clifford group 136.In these embodiments, circuit identification component 112 can implement(e.g., via processor 106) algorithm 1 and/or lemma 1 defined above toidentify, from the quantum circuits 118 (e.g., from the generatedClifford quantum circuits 134 having a Clifford group 136), a desiredcircuit 130 that matches the quantum circuit representation 116. Inthese embodiments, such a desired circuit 130 can comprise a Cliffordquantum circuit 134 having a Clifford group 136 and a defined number ofCNOT gates 132. For instance, in these embodiments, the desired circuit130 can comprise a Clifford quantum circuit 134 having a Clifford group136 and the least number (e.g., smallest number) of CNOT gates 132(e.g., the least number of CNOT gates 132 relative to the number of CNOTgates 132 in other successively generated Clifford quantum circuits134). Consequently, in these embodiments, it should be appreciated thatcircuit generation component 110 can implement (e.g., via processor 106)algorithm 1 and/or lemma 1 defined above to generate, iteratively, thequantum circuits 118 (e.g., Clifford quantum circuits 134 having aClifford group 136) to minimize a number of CNOT gates 132 in thedesired circuit 130. For instance, in these embodiments, circuitgeneration component 110 can implement (e.g., via processor 106)algorithm 1 and/or lemma 1 defined above to successively generate aClifford quantum circuit 134 having a Clifford group 136 and a definednumber of CNOT gates 132 (e.g., the least number of CNOT gates 132relative to the number of CNOT gates 132 in other successively generatedClifford quantum circuits 134).

In some embodiments, quantum circuits 118 described above can comprise aCNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140. Inthese embodiments, circuit generation component 110 can implement (e.g.,via processor 106) algorithm 2 and/or lemma 2 defined above tosuccessively generate the CNOT-Dihedral quantum circuit 138 having aCNOT-Dihedral group 140. In these embodiments, circuit identificationcomponent 112 can implement (e.g., via processor 106) algorithm 2 and/orlemma 2 defined above to identify, from the quantum circuits 118 (e.g.,from the generated CNOT-Dihedral quantum circuits 138 having aCNOT-Dihedral group 140), a desired circuit 130 that matches the quantumcircuit representation 116. In these embodiments, such a desired circuit130 can comprise a CNOT-Dihedral quantum circuit 138 having aCNOT-Dihedral group 140 and a defined number of CNOT gates 132. Forinstance, in these embodiments, the desired circuit 130 can comprise aCNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140 andthe least number (e.g., smallest number) of CNOT gates 132 (e.g., theleast number of CNOT gates 132 relative to the number of CNOT gates 132in other successively generated CNOT-Dihedral quantum circuits 138).Consequently, in these embodiments, it should be appreciated thatcircuit generation component 110 can implement (e.g., via processor 106)algorithm 2 and/or lemma 2 defined above to generate, iteratively, thequantum circuits 118 (e.g., CNOT-Dihedral quantum circuits 138 having aCNOT-Dihedral group 140) to minimize a number of CNOT gates 132 in thedesired circuit 130. For instance, in these embodiments, circuitgeneration component 110 can implement (e.g., via processor 106)algorithm 2 and/or lemma 2 defined above to successively generate aCNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140 and adefined number of CNOT gates 132 (e.g., the least number of CNOT gates132 relative to the number of CNOT gates 132 in other successivelygenerated CNOT-Dihedral quantum circuits 138).

Turning now to the example embodiment illustrated in FIG. 2 . Quantumcircuit synthesis system 102 of system 200 depicted in FIG. 2 cancomprise an example, non-limiting alternative embodiment of quantumcircuit synthesis system 102 of system 100 described above and depictedin FIG. 1 , where quantum circuit synthesis system 102 of system 200 canfurther comprise a second circuit generation component 202 and/or anapplication component 204.

In some embodiments, based on interface component 108 (e.g., an API, aREST API, a GUI, etc.) receiving a quantum circuit representation 116,circuit generation component 110 can implement (e.g., via processor 106)one or more of the algorithms and/or lemmas described above to generate,during a first iteration 206, a first set of quantum circuits 208comprising 2-qubit gates 120. In these embodiments, second circuitgeneration component 202 can implement (e.g., via processor 106) one ormore of the algorithms and/or lemmas described above to generate, duringa second iteration 210, a second set of quantum circuits 212 by adding a2-qubit gate 124 to the first set of quantum circuits 208 such that thesecond set of quantum circuits 212 are selected to a redundant operation226 use the 2-qubit gate 124 without introducing redundant operation 226to that of the first set of quantum circuits 208. In these embodiments,circuit identification component 112 can implement (e.g., via processor106) one or more of the algorithms and/or lemmas described above toidentify, from the first set of quantum circuits 208 and the second setof quantum circuits 212, a desired circuit 130 that matches the quantumcircuit representation 116. For example, in these embodiments, circuitidentification component 112 can identify, from the first set of quantumcircuits 208 and the second set of quantum circuits 212, such a desiredcircuit 130 that matches the quantum circuit representation 116, wherethe desired circuit 130 can comprise a defined number of CNOT gates 132.For instance, in these embodiments, the added 2-qubit gate 124 describedabove can comprise CNOT gates 132 that represent operations 128 distinctfrom previous operations 142 (e.g., not redundant operations), and assuch, the desired circuit 130 can comprise the least number (e.g.,smallest number) of CNOT gates 132 (e.g., the least number of CNOT gates132 relative to the number of CNOT gates 132 in other successivelygenerated quantum circuits 118 in the first set of quantum circuits 208and/or the second set of quantum circuits 212).

In some embodiments, the first set of quantum circuits 208 and thesecond set of quantum circuits 212 described above can respectivelycomprise a first set of Clifford quantum circuits 214 having a Cliffordgroup 136 and a second set of Clifford quantum circuits 216 having aClifford group 136, where the first set of Clifford quantum circuits 214and the second set of Clifford quantum circuits 216 can respectivelycomprise Clifford quantum circuits 134 having a Clifford group 136. Inthese embodiments, circuit generation component 110 and second circuitgeneration component 202 can implement (e.g., via processor 106)algorithm 1 and/or lemma 1 defined above to successively generate thefirst set of Clifford quantum circuits 214 having a Clifford group 136and the second set of Clifford quantum circuits 216 having a Cliffordgroup 136. In these embodiments, circuit identification component 112can implement (e.g., via processor 106) algorithm 1 and/or lemma 1defined above to identify, from the first set of quantum circuits 208and the second set of quantum circuits 212 (e.g., from the first set ofClifford quantum circuits 214 having a Clifford group 136 and the secondset of Clifford quantum circuits 216 having a Clifford group 136), adesired circuit 130 that matches the quantum circuit representation 116.In these embodiments, such a desired circuit 130 can comprise a Cliffordquantum circuit 134 having a Clifford group 136 and a defined number ofCNOT gates 132. For instance, in these embodiments, the desired circuit130 can comprise a Clifford quantum circuit 134 having a Clifford group136 and the least number (e.g., smallest number) of CNOT gates 132(e.g., the least number of CNOT gates 132 relative to the number of CNOTgates 132 in other successively generated Clifford quantum circuits 134in the first set of Clifford quantum circuits 214 and the second set ofClifford quantum circuits 216). Consequently, in these embodiments, itshould be appreciated that circuit generation component 110 canimplement (e.g., via processor 106) algorithm 1 and/or lemma 1 definedabove to generate, iteratively, the first set of quantum circuits 208and the second set of quantum circuits 212 (e.g., the first set ofClifford quantum circuits 214 having a Clifford group 136 and the secondset of Clifford quantum circuits 216 having a Clifford group 136) tominimize a number of CNOT gates 132 in the desired circuit 130. Forinstance, in these embodiments, circuit generation component 110 canimplement (e.g., via processor 106) algorithm 1 and/or lemma 1 definedabove to successively generate a Clifford quantum circuit 134 having aClifford group 136 and a defined number of CNOT gates 132 (e.g., theleast number of CNOT gates 132 relative to the number of CNOT gates 132in other successively generated Clifford quantum circuits 134 in thefirst set of Clifford quantum circuits 214 and the second set ofClifford quantum circuits 216).

In some embodiments, the first set of quantum circuits 208 and thesecond set of quantum circuits 212 described above can respectivelycomprise a first set of CNOT-Dihedral quantum circuits 218 having aCNOT-Dihedral group 140 and a second set of CNOT-Dihedral quantumcircuits 220 having a CNOT-Dihedral group 140, where the first set ofCNOT-Dihedral quantum circuits 218 and the second set of CNOT-Dihedralquantum circuits 220 can respectively comprise CNOT-Dihedral quantumcircuits 138 having a CNOT-Dihedral group 140. In these embodiments,circuit generation component 110 and second circuit generation component202 can implement (e.g., via processor 106) algorithm 2 and/or lemma 2defined above to successively generate the first set of CNOT-Dihedralquantum circuits 218 having a CNOT-Dihedral group 140 and the second setof CNOT-Dihedral quantum circuits 220 having a CNOT-Dihedral group 140.In these embodiments, circuit identification component 112 can implement(e.g., via processor 106) algorithm 2 and/or lemma 2 defined above toidentify, from the first set of quantum circuits 208 and the second setof quantum circuits 212 (e.g., from the first set of CNOT-Dihedralquantum circuits 218 having a CNOT-Dihedral group 140 and the second setof CNOT-Dihedral quantum circuits 220 having a CNOT-Dihedral group 140),a desired circuit 130 that matches the quantum circuit representation116. In these embodiments, such a desired circuit 130 can comprise aCNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140 and adefined number of CNOT gates. For instance, in these embodiments, thedesired circuit 130 can comprise a CNOT-Dihedral quantum circuit 138having a CNOT-Dihedral group 140 and the least number (e.g., smallestnumber) of CNOT gates 132 (e.g., the least number of CNOT gates 132relative to the number of CNOT gates 132 in other successively generatedCNOT-Dihedral quantum circuits 138 in the first set of CNOT-Dihedralquantum circuits 218 and the second set of CNOT-Dihedral quantumcircuits 220). Consequently, in these embodiments, it should beappreciated that circuit generation component 110 can implement (e.g.,via processor 106) algorithm 2 and/or lemma 2 defined above to generate,iteratively, the first set of quantum circuits 208 and the second set ofquantum circuits 212 (e.g., the first set of CNOT-Dihedral quantumcircuits 218 having a CNOT-Dihedral group 140 and the second set ofCNOT-Dihedral quantum circuits 220 having a CNOT-Dihedral group 140) tominimize a number of CNOT gates 132 in the desired circuit 130. Forinstance, in these embodiments, circuit generation component 110 canimplement (e.g., via processor 106) algorithm 2 and/or lemma 2 definedabove to successively generate a CNOT-Dihedral quantum circuit 138having a CNOT-Dihedral group 140 and a defined number of CNOT gates 132(e.g., the least number of CNOT gates 132 relative to the number of CNOTgates 132 in other successively generated CNOT-Dihedral quantum circuits138 in the first set of CNOT-Dihedral quantum circuits 218 and thesecond set of CNOT-Dihedral quantum circuits 220).

Continuing now with the example embodiment illustrated in FIG. 2 .Application component 204 can perform randomized benchmarking on adefined number of qubits 222 in a quantum device 224 based on (e.g.,using) the desired circuit 130 described above that can be identified bycircuit identification component 112. In an example embodiment,application component 204 can comprise a classical computer (e.g.,computer 812, etc.) that can execute (e.g., via processor 106,processing unit 814, etc.) a randomized benchmarking algorithm toperform randomized benchmarking on a limited number of qubits 222 (e.g.,more than 2 qubits, for instance, 3 qubits, 4 qubits, 5 qubits, etc.) ina quantum device 224 (e.g., a quantum processor, a quantum circuit, asuperconducting circuit, etc.) based on (e.g., using) the desiredcircuit 130 described above. In this example embodiment, such a randombenchmarking algorithm can comprise an algorithm that provides anefficient and reliable experimental estimation of an average error-ratefor a set of quantum gate operations by running sequences of randomgates from the Clifford group 136 or CNOT-Dihedral group 140 that shouldreturn the qubits 222 in the quantum device 224 to the initial state. Inthis example embodiment, by performing randomized benchmarking using thedesired circuit 130 described above (e.g., a Clifford quantum circuit134 having a Clifford group 136 or a CNOT-Dihedral quantum circuit 138having a CNOT-Dihedral group 140) that can comprise the least number ofCNOT gates 132 relative to other successively generated quantum circuits118, quantum circuit synthesis system 102 and/or components thereof(e.g., circuit generation component 110, circuit identificationcomponent 112, second circuit generation component 202, applicationcomponent 204, etc.) can thereby facilitate improved efficiency,improved performance, and/or reduced computational costs associated withapplication component 204, processor 106, system 200, and/or quantumcircuit synthesis system 102 in performing the randomized benchmarkingbased on (e.g., using) the desired circuit 130.

Quantum circuit synthesis system 102 can be associated with varioustechnologies. For example, quantum circuit synthesis system 102 can beassociated with quantum computing technologies, quantum circuittechnologies, quantum circuit synthesis technologies, quantum algorithmtechnologies, quantum computing simulation technologies, quantumhardware and/or software technologies, quantum hardware and/or softwarecalibration technologies, machine learning technologies, artificialintelligence technologies, cloud computing technologies, and/or othertechnologies.

Quantum circuit synthesis system 102 can provide technical improvementsto systems, devices, components, operational steps, and/or processingsteps associated with the various technologies identified above. Forexample, quantum circuit synthesis system 102 can: receive a quantumcircuit representation 116; generate, iteratively, quantum circuits 118(also referred to herein as circuits 118) from 1 to N two-qubit gates120, wherein at least one or more iterations 122 (e.g., 1, 2, . . . , N,where N denotes a total quantity) adds a single two-qubit gate 124 tocircuits 118 from a previous iteration 126 based on using added single2-qubit gates 124 that represent operations 128 distinct from previousoperations 142 relative to previous iterations 126; and/or identify,from the quantum circuits 118, a desired circuit 130 that matches thequantum circuit representation 116. An advantage of quantum circuitsynthesis system 102 is that it can minimize the number of CNOT gates132 in the desired circuit 130, which can comprise a Clifford quantumcircuit 134 having a Clifford group 136 and/or a CNOT-Dihedral quantumcircuit 138 having a CNOT-Dihedral group 140.

In the example above, by successively generating quantum circuits 118such that desired circuit 130 has a minimal number of CNOT gates 132 asdescribed herein, quantum circuit synthesis system 102 can therebyfacilitate improved application of such desired circuit 130, as it canbe efficiently generated, manipulated, and/or implemented using aclassical computer to perform a variety of tasks. In this example, asquantum circuits 118 and desired circuit 130 can comprise Cliffordquantum circuit(s) 134 having a Clifford group 136 and/or CNOT-Dihedralquantum circuit(s) 138 having a CNOT-Dihedral group, it should beappreciated that quantum circuit synthesis system 102 can therebyfacilitate improved application of such Clifford quantum circuit(s) 134and/or CNOT-Dihedral quantum circuit(s) 138, as they can be efficientlygenerated, manipulated, and/or implemented using a classical computer toperform a variety of tasks. For instance, by successively generating theClifford quantum circuit 134 and/or the CNOT-Dihedral quantum circuit138 described above such that they have a minimal number of CNOT gates132, quantum circuit synthesis system 102 can thereby enable use of suchquantum circuits to efficiently: construct quantum error correctioncodes called stabilizer codes; optimize universal sets of gates (e.g.,Clifford+T and/or Clifford+CS (controlled-S)) and simulation (e.g.,using the Gottesman-Knill Theorem for the Clifford group 136); and/orbenchmark a quantum device 224 using a random benchmarking algorithm(e.g., as described above with reference to FIG. 2 ).

In another example, by facilitating efficient generation, manipulation,and/or implementation of the Clifford quantum circuit 134 and/or theCNOT-Dihedral quantum circuit 138 described above that have a minimalnumber of CNOT gates 132, quantum circuit synthesis system 102 canthereby reduce the workload of a component, a processor, and/or a systemused to generate, manipulate, and/or implement such quantum circuits. Inthis example, by reducing the workload of such a component, processor,and/or system used to generate, manipulate, and/or implement theClifford quantum circuit 134 and/or the CNOT-Dihedral quantum circuit138 described above, quantum circuit synthesis system 102 can therebyfacilitate improved efficiency, improved performance, and/or reducedcomputational costs associated with such a component (e.g., applicationcomponent 204), processor (e.g., processor 106, processing unit 814,etc.), and/or a system (e.g., system 100, system 200, operatingenvironment 800, computer 812, etc.) used to generate, manipulate,and/or implement such quantum circuits.

Quantum circuit synthesis system 102 can provide technical improvementsto a processing unit (e.g., processor 106, processing unit 814, aquantum processor, etc.) associated with a classical computing deviceand/or a quantum computing device (e.g., a quantum processor, quantumhardware, superconducting circuit, etc.) associated with quantum circuitsynthesis system 102. For example, as described above, by successivelygenerating a Clifford quantum circuit 134 having a Clifford group 136and/or a CNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group140 such that they have a minimal number of CNOT gates 132 as describedherein, quantum circuit synthesis system 102 can thereby facilitateefficient generation, manipulation, and/or implementation of suchquantum circuits using a classical computer to perform a variety oftasks. In this example, by facilitating efficient generation,manipulation, and/or implementation of the Clifford quantum circuit 134and/or the CNOT-Dihedral quantum circuit 138 described above that have aminimal number of CNOT gates 132, quantum circuit synthesis system 102can thereby reduce the workload of a processor (e.g., processor 106,processing unit 814, etc.) used to generate, manipulate, and/orimplement such quantum circuits described above.

In the example above, by reducing the workload of such a processor(e.g., processor 106, processing unit 814, etc.) used to generate,manipulate, and/or implement the Clifford quantum circuit 134 and/or theCNOT-Dihedral quantum circuit 138 described above, quantum circuitsynthesis system 102 can thereby facilitate improved efficiency,improved performance, and/or reduced computational costs associated withsuch a processor in performing such operation(s). For instance, quantumcircuit synthesis system 102 can enable improved efficiency, improvedperformance, and/or reduced computational costs associated withprocessor 106 that can be employed by quantum circuit synthesis system102 and/or components thereof to successively generate the Cliffordquantum circuit 134 and/or the CNOT-Dihedral quantum circuit 138described above in accordance with one or more embodiments of thesubject disclosure described herein. In another example, quantum circuitsynthesis system 102 can enable improved efficiency, improvedperformance, and/or reduced computational costs associated with aprocessing unit (e.g., processor 106) that can be employed byapplication component 204 to perform random benchmarking on a quantumdevice 224 as described above with reference to FIG. 2 .

Based on such successive generation of the Clifford quantum circuit 134and/or the CNOT-Dihedral quantum circuit 138 described above that have aminimal number of CNOT gates 132, a practical application of quantumcircuit synthesis system 102 is that it can be used to generate andimplement one or both of such quantum circuits to benchmark a quantumdevice 224 that can then be used to compute one or more solutions (e.g.,heuristic(s), etc.) to a variety of problems ranging in complexity(e.g., an estimation problem, an optimization problem, etc.) in avariety of domains (e.g., finance, chemistry, medicine, etc.). Forexample, a practical application of quantum circuit synthesis system 102is that it can be deployed using a classical computer (e.g., computer812) to generate and implement one or both of the Clifford quantumcircuit 134 and/or the CNOT-Dihedral quantum circuit 138 to benchmark aquantum device 224 that can then be used to compute a solution (e.g., aheuristic) to an optimization problem and/or an estimation problem inthe domain of chemistry, medicine, and/or finance, where such a solutioncan be used to engineer, for instance, a new chemical compound, a newmedication, and/or a new options pricing system and/or method.

It should be appreciated that quantum circuit synthesis system 102provides a new approach driven by relatively new quantum computingtechnologies. For example, quantum circuit synthesis system 102 providesa new approach to efficiently generate a Clifford quantum circuit 134having a Clifford group 136 and/or a CNOT-Dihedral quantum circuit 138having a CNOT-Dihedral group 140, where such quantum circuits have aminimal number of CNOT gates 132, thereby improving the synthesis and/orapplication of such quantum circuits. In this example, such a newapproach that improves the synthesis and/or application of the Cliffordquantum circuit 134 and/or the CNOT-Dihedral quantum circuit 138 canenable faster and more efficient benchmarking of a quantum device 224that can then be used to compute a solution to such an optimizationand/or estimation problem as described above.

Quantum circuit synthesis system 102 can employ hardware or software tosolve problems that are highly technical in nature, that are notabstract and that cannot be performed as a set of mental acts by ahuman. In some embodiments, one or more of the processes describedherein can be performed by one or more specialized computers (e.g., aspecialized processing unit, a specialized classical computer, aspecialized quantum computer, etc.) to execute defined tasks related tothe various technologies identified above. Quantum circuit synthesissystem 102 and/or components thereof, can be employed to solve newproblems that arise through advancements in technologies mentionedabove, employment of quantum computing systems, cloud computing systems,computer architecture, and/or another technology.

It is to be appreciated that quantum circuit synthesis system 102 canutilize various combinations of electrical components, mechanicalcomponents, and circuitry that cannot be replicated in the mind of ahuman or performed by a human, as the various operations that can beexecuted by quantum circuit synthesis system 102 and/or componentsthereof as described herein are operations that are greater than thecapability of a human mind. For instance, the amount of data processed,the speed of processing such data, or the types of data processed byquantum circuit synthesis system 102 over a certain period of time canbe greater, faster, or different than the amount, speed, or data typethat can be processed by a human mind over the same period of time.

According to several embodiments, quantum circuit synthesis system 102can also be fully operational towards performing one or more otherfunctions (e.g., fully powered on, fully executed, etc.) while alsoperforming the various operations described herein. It should beappreciated that such simultaneous multi-operational execution is beyondthe capability of a human mind. It should also be appreciated thatquantum circuit synthesis system 102 can include information that isimpossible to obtain manually by an entity, such as a human user. Forexample, the type, amount, and/or variety of information included inquantum circuit synthesis system 102, interface component 108, circuitgeneration component 110, circuit identification component 112, secondcircuit generation component 202, and/or application component 204 canbe more complex than information obtained manually by a human user.

FIG. 3 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 300 that can facilitate synthesis of aquantum circuit in accordance with one or more embodiments describedherein. Repetitive description of like elements and/or processesemployed in respective embodiments is omitted for sake of brevity.

At 302, computer-implemented method 300 can comprise generating, by asystem (e.g., via quantum circuit synthesis system 102 and/or circuitgeneration component 110) operatively coupled to a processor (e.g.,processor 106), iteratively, quantum circuits (e.g., quantum circuits118 comprising Clifford quantum circuits 134 having a Clifford group 136and/or CNOT-Dihedral quantum circuits 138 having a CNOT-Dihedral group140) from 1 to N two-qubit gates (e.g., two-qubit gates 120), where atleast one or more iterations (1, 2, . . . , N)(e.g., at least one ormore iterations 122) adds a single two-qubit gate (e.g., singletwo-qubit gate 124) to circuits (e.g., quantum circuits 118) from aprevious iteration (e.g., previous iteration 126) based on using addedsingle 2-qubit gates (e.g., added single 2-qubit gates 124) thatrepresent operations (e.g., operations 128) distinct from previousoperations (e.g., previous operations 142) relative to previousiterations (e.g., previous iterations 126).

At 304, computer-implemented method 300 can comprise identifying, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitidentification component 112), from the quantum circuits, a desiredcircuit (e.g., desired circuit 130 comprising a Clifford quantum circuit134 having a Clifford group 136 and a minimal number of CNOT gates 132or a CNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140and a minimal number of CNOT gates 132) that matches a quantum circuitrepresentation (e.g., quantum circuit representation 116).

FIG. 4 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 300 that can facilitate synthesis of aquantum circuit in accordance with one or more embodiments describedherein. Repetitive description of like elements and/or processesemployed in respective embodiments is omitted for sake of brevity.

At 402, computer-implemented method 400 can comprise receiving, by asystem (e.g., via quantum circuit synthesis system 102 and/or interfacecomponent 108) operatively coupled to a processor (e.g., processor 106,a quantum processor, etc.), a quantum circuit representation (e.g.,quantum circuit representation 116).

At 404, computer-implemented method 400 can comprise generating, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitgeneration component 110), iteratively, quantum circuits (e.g., quantumcircuits 118 comprising Clifford quantum circuits 134 having a Cliffordgroup 136 and/or CNOT-Dihedral quantum circuits 138 having aCNOT-Dihedral group 140) from 1 to N two-qubit gates (e.g., two-qubitgates 120), wherein at least one or more iterations (1, 2, . . . ,N)(e.g., at least one or more iterations 122) adds a single two-qubitgate (e.g., single two-qubit gate 124) to circuits (e.g., quantumcircuits 118) from a previous iteration (e.g., previous iteration 126)based on using added single 2-qubit gates (e.g., added single 2-qubitgates 124) that represent operations (e.g., operations 128) distinctfrom previous operations (e.g., previous operations 142) relative toprevious iterations (e.g., previous iterations 126).

At 406, computer-implemented method 400 can comprise identifying, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitidentification component 112), from the quantum circuits, a desiredcircuit (e.g., desired circuit 130 comprising a Clifford quantum circuit134 having a Clifford group 136 and a minimal number of CNOT gates 132or a CNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140and a minimal number of CNOT gates 132) that matches the quantum circuitrepresentation.

At 408, computer-implemented method 400 can comprise generating, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitgeneration component 110), iteratively, the quantum circuits to minimizea number of CNOT gates (e.g., CNOT gates 132) in the desired circuit.

At 410, computer-implemented method 400 can comprise performing, by thesystem (e.g., via quantum circuit synthesis system 102 and/orapplication component 204), randomized benchmarking (e.g., via employinga randomized benchmarking algorithm as described above with reference toFIG. 2 ) on a defined number of qubits (e.g., qubits 222 comprising morethan 2 qubits, for instance, 3 qubits, 4 qubits, 5 qubits, etc.) in aquantum device (e.g., quantum device 224) based on (e.g., using) thedesired circuit, wherein the desired circuit comprises a defined numberof CNOT gates (e.g., CNOT gates 132) and a Clifford quantum circuithaving a Clifford group or a CNOT-Dihedral quantum circuit having aCNOT-Dihedral group (e.g., a Clifford quantum circuit 134 having aClifford group 136 or a CNOT-Dihedral quantum circuit 138 having aCNOT-Dihedral group 140), thereby facilitating at least one of improvedefficiency, improved performance, or reduced computational costsassociated with at least one of the processor (e.g., processor 106) orthe system (e.g., system 100, quantum circuit synthesis system 102,operating environment 800, computer 812, etc.) in performing therandomized benchmarking based on the desired circuit 130.

FIG. 5 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 500 that can facilitate synthesis of aquantum circuit in accordance with one or more embodiments describedherein. Repetitive description of like elements and/or processesemployed in respective embodiments is omitted for sake of brevity.

At 502, computer-implemented method 500 can comprise generating, by asystem (e.g., via quantum circuit synthesis system 102 and/or circuitgeneration component 110) operatively coupled to a processor (e.g.,processor 106), during a first iteration (e.g., first iteration 206), afirst set of quantum circuits (e.g., first set of quantum circuits 208comprising a first set of Clifford quantum circuits 214 having aClifford group 136 and/or a first set of CNOT-Dihedral quantum circuits218 having a CNOT-Dihedral group 140) comprising 2-qubit gates (e.g.,two-qubit gates 124).

At 504, computer-implemented method 500 can comprise generating, by thesystem (e.g., via quantum circuit synthesis system 102 and/or secondcircuit generation component 202), during a second iteration (e.g.,second iteration 210), a second set of quantum circuits (e.g., secondset of quantum circuits 212 comprising a second set of Clifford quantumcircuits 216 having a Clifford group 136 and/or a second set ofCNOT-Dihedral quantum circuits 220 having a CNOT-Dihedral group 140) byadding a 2-qubit gate (e.g., single 2-qubit gate 124) to the first setof quantum circuits such that the second set of quantum circuits areselected to a redundant operation (e.g., redundant operation 226) usethe 2-qubit gate without introducing the redundant operation to that ofthe first set of quantum circuits.

At 506, computer-implemented method 500 can comprise identifying, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitidentification component 112), from the first set of quantum circuitsand the second set of quantum circuits, a desired circuit (e.g., desiredcircuit 130 comprising a Clifford quantum circuit 134 having a Cliffordgroup 136 and a minimal number of CNOT gates 132 or a CNOT-Dihedralquantum circuit 138 having a CNOT-Dihedral group 140 and a minimalnumber of CNOT gates 132) that matches a quantum circuit representation(e.g., quantum circuit representation 116).

FIG. 6 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 600 that can facilitate synthesis of aquantum circuit in accordance with one or more embodiments describedherein. Repetitive description of like elements and/or processesemployed in respective embodiments is omitted for sake of brevity.

At 602, computer-implemented method 600 can comprise receiving, by asystem (e.g., via quantum circuit synthesis system 102 and/or interfacecomponent 108) operatively coupled to a processor (e.g., processor 106,a quantum processor, etc.), a quantum circuit representation (e.g.,quantum circuit representation 116).

At 604, computer-implemented method 600 can comprise generating, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitgeneration component 110), during a first iteration (e.g., firstiteration 206), a first set of quantum circuits (e.g., first set ofquantum circuits 208 comprising a first set of Clifford quantum circuits214 having a Clifford group 136 and/or a first set of CNOT-Dihedralquantum circuits 218 having a CNOT-Dihedral group 140) comprising2-qubit gates (e.g., two-qubit gates 124).

At 606, computer-implemented method 600 can comprise generating, by thesystem (e.g., via quantum circuit synthesis system 102 and/or secondcircuit generation component 202), during a second iteration (e.g.,second iteration 210), a second set of quantum circuits (e.g., secondset of quantum circuits 212 comprising a second set of Clifford quantumcircuits 216 having a Clifford group 136 and/or a second set ofCNOT-Dihedral quantum circuits 220 having a CNOT-Dihedral group 140) byadding a 2-qubit gate (e.g., single 2-qubit gate 124) to the first setof quantum circuits such that the second set of quantum circuits areselected to a redundant operation (e.g., redundant operation 226) usethe 2-qubit gate without introducing the redundant operation to that ofthe first set of quantum circuits.

At 608, computer-implemented method 600 can comprise identifying, by thesystem (e.g., via quantum circuit synthesis system 102 and/or circuitidentification component 112), from the first set of quantum circuitsand the second set of quantum circuits, a desired circuit (e.g., desiredcircuit 130 comprising a Clifford quantum circuit 134 having a Cliffordgroup 136 and a minimal number of CNOT gates 132 or a CNOT-Dihedralquantum circuit 138 having a CNOT-Dihedral group 140 and a minimalnumber of CNOT gates 132) that matches the quantum circuitrepresentation.

At 610, computer-implemented method 600 can comprise generating, by thesystem (e.g., via quantum circuit synthesis system 102, circuitgeneration component 110, and/or second circuit generation component202), iteratively, the first set of quantum circuits and the second setof quantum circuits to minimize a number of CNOT gates (e.g., CNOT gates132) in the desired circuit.

At 612, computer-implemented method 600 can comprise performing, by thesystem (e.g., via quantum circuit synthesis system 102, applicationcomponent 204) randomized benchmarking (e.g., via employing a randomizedbenchmarking algorithm as described above with reference to FIG. 2 ) ona defined number of qubits (e.g., qubits 222 comprising more than 2qubits, for instance, 3 qubits, 4 qubits, 5 qubits, etc.) in a quantumdevice (e.g., quantum device 224) based on (e.g., using) the desiredcircuit, wherein the desired circuit comprises a defined number of CNOTgates (e.g., CNOT gates 132) and a Clifford quantum circuit having aClifford group or a CNOT-Dihedral quantum circuit having a CNOT-Dihedralgroup (e.g., a Clifford quantum circuit 134 having a Clifford group 136or a CNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group140), thereby facilitating at least one of improved efficiency, improvedperformance, or reduced computational costs associated with at least oneof the processor (e.g., processor 106) or the system (e.g., system 200,quantum circuit synthesis system 102, operating environment 800,computer 812, etc.) in performing the randomized benchmarking based onthe desired circuit 130.

FIG. 7 illustrates a flow diagram of an example, non-limitingcomputer-implemented method 700 that can facilitate synthesis of aquantum circuit in accordance with one or more embodiments describedherein. Repetitive description of like elements and/or processesemployed in respective embodiments is omitted for sake of brevity.

At 702, computer-implemented method 700 can comprise receiving (e.g.,via quantum circuit synthesis system 102 and/or interface component 108)a quantum circuit representation (e.g., quantum circuit representation116).

At 704, computer-implemented method 700 can comprise generating (e.g.,via quantum circuit synthesis system 102 and/or circuit generationcomponent 110), iteratively, quantum circuits (e.g., quantum circuits118 comprising Clifford quantum circuits 134 having a Clifford group 136and/or CNOT-Dihedral quantum circuits 138 having a CNOT-Dihedral group140) from 1 to N two-qubit gates (e.g., two-qubit gates 120) such that,at each iteration (e.g., iteration 122), a single two-qubit gate (e.g.,single two-qubit gate 124) can be added to quantum circuits generated ina previous iteration (e.g., previous iteration 126).

At 706, computer-implemented method 700 can comprise determining (e.g.,via quantum circuit synthesis system 102 and/or circuit generationcomponent 110) whether the single two-qubit gate that can be added tothe quantum circuits generated in a previous iteration (e.g., previousiteration 126) introduce a redundant operation (e.g., an operation 128that is redundant with respect to previous operations 142) relative tosuch previously generated quantum circuits.

If it is determined at 706 that the single two-qubit gate that can beadded to the quantum circuits generated in a previous iteration (e.g.,previous iteration 126) does not introduce a redundant operation (e.g.,an operation 128 that is redundant with respect to previous operations142) relative to such previously generated quantum circuits, at 708,computer-implemented method 700 can comprise adding (e.g., via quantumcircuit synthesis system 102 and/or circuit generation component 110)the single two-qubit gate to the quantum circuits generated in aprevious iteration (e.g., previous iteration 126) and returning tooperation 704. In various embodiments, operations 704, 706, and 708 canbe repeated until it is determined at 706 that the single two-qubit gatethat can be added to the quantum circuits generated in a previousiteration (e.g., previous iteration 126) does not introduce a redundantoperation (e.g., an operation 128 that is redundant with respect toprevious operations 142) relative to such previously generated quantumcircuits.

If it is determined at 706 that the single two-qubit gate that can beadded to the quantum circuits generated in a previous iteration (e.g.,previous iteration 126) does introduce a redundant operation (e.g., anoperation 128 that is redundant with respect to previous operations 142)relative to such previously generated quantum circuits, at 710,computer-implemented method 700 can comprise identifying (e.g., viaquantum circuit synthesis system 102 and/or circuit identificationcomponent 112), from the quantum circuits, a desired circuit (e.g.,desired circuit 130 comprising a Clifford quantum circuit 134 having aClifford group 136 and a minimal number of CNOT gates 132 or aCNOT-Dihedral quantum circuit 138 having a CNOT-Dihedral group 140 and aminimal number of CNOT gates 132) that matches the quantum circuitrepresentation.

At 712, computer-implemented method 700 can comprise performing (e.g.,via quantum circuit synthesis system 102 and/or application component204) a quantum computing based task 714 using the desired circuit (e.g.,performing randomized benchmarking on qubits 222 in a quantum device 224as described above with reference to FIG. 2 ).

For simplicity of explanation, the computer-implemented methodologiesare depicted and described as a series of acts. It is to be understoodand appreciated that the subject innovation is not limited by the actsillustrated and/or by the order of acts, for example acts can occur invarious orders and/or concurrently, and with other acts not presentedand described herein. Furthermore, not all illustrated acts can berequired to implement the computer-implemented methodologies inaccordance with the disclosed subject matter. In addition, those skilledin the art will understand and appreciate that the computer-implementedmethodologies could alternatively be represented as a series ofinterrelated states via a state diagram or events. Additionally, itshould be further appreciated that the computer-implementedmethodologies disclosed hereinafter and throughout this specificationare capable of being stored on an article of manufacture to facilitatetransporting and transferring such computer-implemented methodologies tocomputers. The term article of manufacture, as used herein, is intendedto encompass a computer program accessible from any computer-readabledevice or storage media.

In order to provide a context for the various aspects of the disclosedsubject matter, FIG. 8 as well as the following discussion are intendedto provide a general description of a suitable environment in which thevarious aspects of the disclosed subject matter can be implemented. FIG.8 illustrates a block diagram of an example, non-limiting operatingenvironment in which one or more embodiments described herein can befacilitated. Repetitive description of like elements employed in otherembodiments described herein is omitted for sake of brevity.

With reference to FIG. 8 , a suitable operating environment 800 forimplementing various aspects of this disclosure can also include acomputer 812. The computer 812 can also include a processing unit 814, asystem memory 816, and a system bus 818. The system bus 818 couplessystem components including, but not limited to, the system memory 816to the processing unit 814. The processing unit 814 can be any ofvarious available processors. Dual microprocessors and othermultiprocessor architectures also can be employed as the processing unit814. The system bus 818 can be any of several types of bus structure(s)including the memory bus or memory controller, a peripheral bus orexternal bus, and/or a local bus using any variety of available busarchitectures including, but not limited to, Industrial StandardArchitecture (ISA), Micro-Channel Architecture (MSA), Extended ISA(EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB),Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus(USB), Advanced Graphics Port (AGP), Firewire (IEEE 1394), and SmallComputer Systems Interface (SCSI).

The system memory 816 can also include volatile memory 820 andnonvolatile memory 822. The basic input/output system (BIOS), containingthe basic routines to transfer information between elements within thecomputer 812, such as during start-up, is stored in nonvolatile memory822. Computer 812 can also include removable/non-removable,volatile/non-volatile computer storage media. FIG. 8 illustrates, forexample, a disk storage 824. Disk storage 824 can also include, but isnot limited to, devices like a magnetic disk drive, floppy disk drive,tape drive, Jaz drive, Zip drive, LS-100 drive, flash memory card, ormemory stick. The disk storage 824 also can include storage mediaseparately or in combination with other storage media. To facilitateconnection of the disk storage 824 to the system bus 818, a removable ornon-removable interface is typically used, such as interface 826. FIG. 8also depicts software that acts as an intermediary between users and thebasic computer resources described in the suitable operating environment800. Such software can also include, for example, an operating system828. Operating system 828, which can be stored on disk storage 824, actsto control and allocate resources of the computer 812.

System applications 830 take advantage of the management of resources byoperating system 828 through program modules 832 and program data 834,e.g., stored either in system memory 816 or on disk storage 824. It isto be appreciated that this disclosure can be implemented with variousoperating systems or combinations of operating systems. A user enterscommands or information into the computer 812 through input device(s)836. Input devices 836 include, but are not limited to, a pointingdevice such as a mouse, trackball, stylus, touch pad, keyboard,microphone, joystick, game pad, satellite dish, scanner, TV tuner card,digital camera, digital video camera, web camera, and the like. Theseand other input devices connect to the processing unit 814 through thesystem bus 818 via interface port(s) 838. Interface port(s) 838 include,for example, a serial port, a parallel port, a game port, and auniversal serial bus (USB). Output device(s) 840 use some of the sametype of ports as input device(s) 836. Thus, for example, a USB port canbe used to provide input to computer 812, and to output information fromcomputer 812 to an output device 840. Output adapter 842 is provided toillustrate that there are some output devices 840 like monitors,speakers, and printers, among other output devices 840, which requirespecial adapters. The output adapters 842 include, by way ofillustration and not limitation, video and sound cards that provide ameans of connection between the output device 840 and the system bus818. It should be noted that other devices and/or systems of devicesprovide both input and output capabilities such as remote computer(s)844.

Computer 812 can operate in a networked environment using logicalconnections to one or more remote computers, such as remote computer(s)844. The remote computer(s) 844 can be a computer, a server, a router, anetwork PC, a workstation, a microprocessor based appliance, a peerdevice or other common network node and the like, and typically can alsoinclude many or all of the elements described relative to computer 812.For purposes of brevity, only a memory storage device 846 is illustratedwith remote computer(s) 844. Remote computer(s) 844 is logicallyconnected to computer 812 through a network interface 848 and thenphysically connected via communication connection 850. Network interface848 encompasses wire and/or wireless communication networks such aslocal-area networks (LAN), wide-area networks (WAN), cellular networks,etc. LAN technologies include Fiber Distributed Data Interface (FDDI),Copper Distributed Data Interface (CDDI), Ethernet, Token Ring and thelike. WAN technologies include, but are not limited to, point-to-pointlinks, circuit switching networks like Integrated Services DigitalNetworks (ISDN) and variations thereon, packet switching networks, andDigital Subscriber Lines (DSL). Communication connection(s) 850 refersto the hardware/software employed to connect the network interface 848to the system bus 818. While communication connection 850 is shown forillustrative clarity inside computer 812, it can also be external tocomputer 812. The hardware/software for connection to the networkinterface 848 can also include, for exemplary purposes only, internaland external technologies such as, modems including regular telephonegrade modems, cable modems and DSL modems, ISDN adapters, and Ethernetcards.

Referring now to FIG. 9 , an illustrative cloud computing environment950 is depicted. As shown, cloud computing environment 950 includes oneor more cloud computing nodes 910 with which local computing devicesused by cloud consumers, such as, for example, personal digitalassistant (PDA) or cellular telephone 954A, desktop computer 954B,laptop computer 954C, and/or automobile computer system 954N maycommunicate. Although not illustrated in FIG. 9 , cloud computing nodes910 can further comprise a quantum platform (e.g., quantum computer,quantum hardware, quantum software, etc.) with which local computingdevices used by cloud consumers can communicate. Nodes 910 maycommunicate with one another. They may be grouped (not shown) physicallyor virtually, in one or more networks, such as Private, Community,Public, or Hybrid clouds as described hereinabove, or a combinationthereof. This allows cloud computing environment 950 to offerinfrastructure, platforms and/or software as services for which a cloudconsumer does not need to maintain resources on a local computingdevice. It is understood that the types of computing devices 954A-Nshown in FIG. 9 are intended to be illustrative only and that computingnodes 910 and cloud computing environment 950 can communicate with anytype of computerized device over any type of network and/or networkaddressable connection (e.g., using a web browser).

Referring now to FIG. 10 , a set of functional abstraction layersprovided by cloud computing environment 950 (FIG. 9 ) is shown. Itshould be understood in advance that the components, layers, andfunctions shown in FIG. 10 are intended to be illustrative only andembodiments of the invention are not limited thereto. As depicted, thefollowing layers and corresponding functions are provided:

Hardware and software layer 1060 includes hardware and softwarecomponents. Examples of hardware components include: mainframes 1061;RISC (Reduced Instruction Set Computer) architecture based servers 1062;servers 1063; blade servers 1064; storage devices 1065; and networks andnetworking components 1066. In some embodiments, software componentsinclude network application server software 1067, database software1068, quantum platform routing software (not illustrated in FIG. 10 ),and/or quantum software (not illustrated in FIG. 10 ).

Virtualization layer 1070 provides an abstraction layer from which thefollowing examples of virtual entities may be provided: virtual servers1071; virtual storage 1072; virtual networks 1073, including virtualprivate networks; virtual applications and operating systems 1074; andvirtual clients 1075.

In one example, management layer 1080 may provide the functionsdescribed below. Resource provisioning 1081 provides dynamic procurementof computing resources and other resources that are utilized to performtasks within the cloud computing environment. Metering and Pricing 1082provide cost tracking as resources are utilized within the cloudcomputing environment, and billing or invoicing for consumption of theseresources. In one example, these resources may include applicationsoftware licenses. Security provides identity verification for cloudconsumers and tasks, as well as protection for data and other resources.User portal 1083 provides access to the cloud computing environment forconsumers and system administrators. Service level management 1084provides cloud computing resource allocation and management such thatrequired service levels are met. Service Level Agreement (SLA) planningand fulfillment 1085 provide pre-arrangement for, and procurement of,cloud computing resources for which a future requirement is anticipatedin accordance with an SLA.

Workloads layer 1090 provides examples of functionality for which thecloud computing environment may be utilized. Non-limiting examples ofworkloads and functions which may be provided from this layer include:mapping and navigation 1091; software development and lifecyclemanagement 1092; virtual classroom education delivery 1093; dataanalytics processing 1094; transaction processing 1095; and quantumcircuit synthesis software 1096.

The present invention may be a system, a method, an apparatus and/or acomputer program product at any possible technical detail level ofintegration. The computer program product can include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention. The computer readable storage medium can be atangible device that can retain and store instructions for use by aninstruction execution device. The computer readable storage medium canbe, for example, but is not limited to, an electronic storage device, amagnetic storage device, an optical storage device, an electromagneticstorage device, a semiconductor storage device, or any suitablecombination of the foregoing. A non-exhaustive list of more specificexamples of the computer readable storage medium can also include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a static random access memory(SRAM), a portable compact disc read-only memory (CD-ROM), a digitalversatile disk (DVD), a memory stick, a floppy disk, a mechanicallyencoded device such as punch-cards or raised structures in a groovehaving instructions recorded thereon, and any suitable combination ofthe foregoing. A computer readable storage medium, as used herein, isnot to be construed as being transitory signals per se, such as radiowaves or other freely propagating electromagnetic waves, electromagneticwaves propagating through a waveguide or other transmission media (e.g.,light pulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network can comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device. Computer readable programinstructions for carrying out operations of the present invention can beassembler instructions, instruction-set-architecture (ISA) instructions,machine instructions, machine dependent instructions, microcode,firmware instructions, state-setting data, configuration data forintegrated circuitry, or either source code or object code written inany combination of one or more programming languages, including anobject oriented programming language such as Smalltalk, C++, or thelike, and procedural programming languages, such as the “C” programminglanguage or similar programming languages. The computer readable programinstructions can execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer can beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection can be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) can execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions. These computer readable programinstructions can be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks. These computer readable program instructions can also be storedin a computer readable storage medium that can direct a computer, aprogrammable data processing apparatus, and/or other devices to functionin a particular manner, such that the computer readable storage mediumhaving instructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks. Thecomputer readable program instructions can also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational acts to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks can occur out of theorder noted in the Figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the subject matter has been described above in the general contextof computer-executable instructions of a computer program product thatruns on a computer and/or computers, those skilled in the art willrecognize that this disclosure also can or can be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, etc. thatperform particular tasks and/or implement particular abstract datatypes. Moreover, those skilled in the art will appreciate that theinventive computer-implemented methods can be practiced with othercomputer system configurations, including single-processor ormultiprocessor computer systems, mini-computing devices, mainframecomputers, as well as computers, hand-held computing devices (e.g., PDA,phone), microprocessor-based or programmable consumer or industrialelectronics, and the like. The illustrated aspects can also be practicedin distributed computing environments in which tasks are performed byremote processing devices that are linked through a communicationsnetwork. However, some, if not all aspects of this disclosure can bepracticed on stand-alone computers. In a distributed computingenvironment, program modules can be located in both local and remotememory storage devices. For example, in one or more embodiments,computer executable components can be executed from memory that caninclude or be comprised of one or more distributed memory units. As usedherein, the term “memory” and “memory unit” are interchangeable.Further, one or more embodiments described herein can execute code ofthe computer executable components in a distributed manner, e.g.,multiple processors combining or working cooperatively to execute codefrom one or more distributed memory units. As used herein, the term“memory” can encompass a single memory or memory unit at one location ormultiple memories or memory units at one or more locations.

As used in this application, the terms “component,” “system,”“platform,” “interface,” and the like, can refer to and/or can include acomputer-related entity or an entity related to an operational machinewith one or more specific functionalities. The entities disclosed hereincan be either hardware, a combination of hardware and software,software, or software in execution. For example, a component can be, butis not limited to being, a process running on a processor, a processor,an object, an executable, a thread of execution, a program, and/or acomputer. By way of illustration, both an application running on aserver and the server can be a component. One or more components canreside within a process and/or thread of execution and a component canbe localized on one computer and/or distributed between two or morecomputers. In another example, respective components can execute fromvarious computer readable media having various data structures storedthereon. The components can communicate via local and/or remoteprocesses such as in accordance with a signal having one or more datapackets (e.g., data from one component interacting with anothercomponent in a local system, distributed system, and/or across a networksuch as the Internet with other systems via the signal). As anotherexample, a component can be an apparatus with specific functionalityprovided by mechanical parts operated by electric or electroniccircuitry, which is operated by a software or firmware applicationexecuted by a processor. In such a case, the processor can be internalor external to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts, wherein the electroniccomponents can include a processor or other means to execute software orfirmware that confers at least in part the functionality of theelectronic components. In an aspect, a component can emulate anelectronic component via a virtual machine, e.g., within a cloudcomputing system.

In addition, the term “or” is intended to mean an inclusive “or” ratherthan an exclusive “or.” That is, unless specified otherwise, or clearfrom context, “X employs A or B” is intended to mean any of the naturalinclusive permutations. That is, if X employs A; X employs B; or Xemploys both A and B, then “X employs A or B” is satisfied under any ofthe foregoing instances. Moreover, articles “a” and “an” as used in thesubject specification and annexed drawings should generally be construedto mean “one or more” unless specified otherwise or clear from contextto be directed to a singular form. As used herein, the terms “example”and/or “exemplary” are utilized to mean serving as an example, instance,or illustration. For the avoidance of doubt, the subject matterdisclosed herein is not limited by such examples. In addition, anyaspect or design described herein as an “example” and/or “exemplary” isnot necessarily to be construed as preferred or advantageous over otheraspects or designs, nor is it meant to preclude equivalent exemplarystructures and techniques known to those of ordinary skill in the art.

As it is employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. Further, processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of user equipment. A processor can also beimplemented as a combination of computing processing units. In thisdisclosure, terms such as “store,” “storage,” “data store,” datastorage,” “database,” and substantially any other information storagecomponent relevant to operation and functionality of a component areutilized to refer to “memory components,” entities embodied in a“memory,” or components comprising a memory. It is to be appreciatedthat memory and/or memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory. By way of illustration, and not limitation,nonvolatile memory can include read only memory (ROM), programmable ROM(PROM), electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g.,ferroelectric RAM (FeRAM). Volatile memory can include RAM, which canact as external cache memory, for example. By way of illustration andnot limitation, RAM is available in many forms such as synchronous RAM(SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rateSDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM),direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), andRambus dynamic RAM (RDRAM). Additionally, the disclosed memorycomponents of systems or computer-implemented methods herein areintended to include, without being limited to including, these and anyother suitable types of memory.

What has been described above include mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describeevery conceivable combination of components or computer-implementedmethods for purposes of describing this disclosure, but one of ordinaryskill in the art can recognize that many further combinations andpermutations of this disclosure are possible. Furthermore, to the extentthat the terms “includes,” “has,” “possesses,” and the like are used inthe detailed description, claims, appendices and drawings such terms areintended to be inclusive in a manner similar to the term “comprising” as“comprising” is interpreted when employed as a transitional word in aclaim.

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

What is claimed is:
 1. A system, comprising: a memory that storescomputer executable components; and a processor that executes thecomputer executable components stored in the memory, wherein thecomputer executable components comprise: a circuit generation componentthat generates, during a first iteration, a first set of quantumcircuits comprising 2-qubit gates; a second circuit generation componentthat generates, during a second iteration, a second set of quantumcircuits by adding a 2-qubit gate to the first set of quantum circuitssuch that the second set of quantum circuits are selected to a redundantoperation use the 2-qubit gate without introducing the redundantoperation to that of the first set of quantum circuits; and a circuitidentification component that identifies, from the first set of quantumcircuits and the second set of quantum circuits, a desired circuit thatmatches a quantum circuit representation.
 2. The system of claim 1,wherein the first set of quantum circuits and the second set of quantumcircuits comprise Clifford quantum circuits having a Clifford group, andwherein the computer executable components further comprise: aninterface component that receives the quantum circuit representation. 3.The system of claim 1, wherein the first set of quantum circuits and thesecond set of quantum circuits comprise CNOT-Dihedral quantum circuitshaving a CNOT-Dihedral group.
 4. The system of claim 1, wherein thecircuit generation component generates, iteratively, the first set ofquantum circuits and the second circuit generation component generates,iteratively, the second set of quantum circuits to minimize a number ofCNOT gates in the desired circuit.
 5. The system of claim 1, wherein thedesired circuit comprises a defined number of CNOT gates and a Cliffordquantum circuit having a Clifford group or a CNOT-Dihedral quantumcircuit having a CNOT-Dihedral group.
 6. The system of claim 5, whereinthe computer executable components further comprise: an applicationcomponent that performs randomized benchmarking on a defined number ofqubits in a quantum device based on the desired circuit, therebyfacilitating at least one of improved efficiency, improved performance,or reduced computational costs associated with at least one of theapplication component, the processor, or the system in performing therandomized benchmarking based on the desired circuit.
 7. Acomputer-implemented method of synthesizing a quantum circuit,comprising: generating, by a system operatively coupled to a processor,during a first iteration, a first set of quantum circuits comprising2-qubit gates; generating, by the system, during a second iteration, asecond set of quantum circuits by adding a 2-qubit gate to the first setof quantum circuits such that the second set of quantum circuits areselected to a redundant operation use the 2-qubit gate withoutintroducing the redundant operation to that of the first set of quantumcircuits; and identifying, by the system, from the first set of quantumcircuits and the second set of quantum circuits, a desired circuit thatmatches a quantum circuit representation.
 8. The computer-implementedmethod of claim 7, wherein the first set of quantum circuits and thesecond set of quantum circuits comprise Clifford quantum circuits havinga Clifford group, and further comprising: receiving, by the system, thequantum circuit representation.
 9. The computer-implemented method ofclaim 7, wherein the first set of quantum circuits and the second set ofquantum circuits comprise CNOT-Dihedral quantum circuits having aCNOT-Dihedral group.
 10. The computer-implemented method of claim 7,further comprising: generating, by the system, iteratively, the firstset of quantum circuits and the second set of quantum circuits tominimize a number of CNOT gates in the desired circuit.
 11. Thecomputer-implemented method of claim 7, wherein the desired circuitcomprises a defined number of CNOT gates and a Clifford quantum circuithaving a Clifford group or a CNOT-Dihedral quantum circuit having aCNOT-Dihedral group.
 12. The computer-implemented method of claim 11,further comprising: performing, by the system, randomized benchmarkingon a defined number of qubits in a quantum device based on the desiredcircuit, thereby facilitating at least one of improved efficiency,improved performance, or reduced computational costs associated with atleast one of the processor or the system in performing the randomizedbenchmarking based on the desired circuit.